Random Walks and Forest Fire Models: Recurrence, Transience and Phase Transitions
Författare
Summary, in English
Random Walk is one of the holy grails of Probability Theory. This literature is born from a simple, nagging
question: what happens to the drunkard’s walk if his steps are no longer equal? The classical theory of random
walks, a cornerstone of probability, offers elegant answers for the simple, symmetric case. But life and many
stochastic models are rarely so uniform.
We step into this gap by studying the Rademacher walk, defined by the sum Sn =
∑n a_iX_i
i=1 , where the
step sizes (ai) are a fixed, deterministic sequence and the Xi are independent Rademacher random variables. Our
central concern is the ancient dichotomy of recurrence and transience: does the path return to a neighbourhood
of the origin infinitely often, or does it wander off forever?
We discover that the answer is a delicate balance between the growth of the step sequence and the geometry of
the path. In one dimension, we find a lower bound of the threshold: if the steps grow like n
α+o(1) for α > 1/2,
the walk is transient, and this bound is exact. But we also construct sequences that grow arbitrarily fast yet still
produce a weakly recurrent walk—a finding that challenges intuition. Moreover, we produce arbitrarily slowly
growing sequences which result in transience. We also extend the work on 2− dimensions and prove similar
results. A parallel study of a forest fire model with delays shows how relaxing standard assumptions can produce
entirely new phenomena, like an “infinite fire.” A parameter of fire spreading time plays a serious role in phase
transition. In addition we find how quickly fires spread.
question: what happens to the drunkard’s walk if his steps are no longer equal? The classical theory of random
walks, a cornerstone of probability, offers elegant answers for the simple, symmetric case. But life and many
stochastic models are rarely so uniform.
We step into this gap by studying the Rademacher walk, defined by the sum Sn =
∑n a_iX_i
i=1 , where the
step sizes (ai) are a fixed, deterministic sequence and the Xi are independent Rademacher random variables. Our
central concern is the ancient dichotomy of recurrence and transience: does the path return to a neighbourhood
of the origin infinitely often, or does it wander off forever?
We discover that the answer is a delicate balance between the growth of the step sequence and the geometry of
the path. In one dimension, we find a lower bound of the threshold: if the steps grow like n
α+o(1) for α > 1/2,
the walk is transient, and this bound is exact. But we also construct sequences that grow arbitrarily fast yet still
produce a weakly recurrent walk—a finding that challenges intuition. Moreover, we produce arbitrarily slowly
growing sequences which result in transience. We also extend the work on 2− dimensions and prove similar
results. A parallel study of a forest fire model with delays shows how relaxing standard assumptions can produce
entirely new phenomena, like an “infinite fire.” A parameter of fire spreading time plays a serious role in phase
transition. In addition we find how quickly fires spread.
Avdelning/ar
Publiceringsår
2026
Språk
Engelska
Fulltext
Dokumenttyp
Doktorsavhandling
Förlag
Centre for Mathematical Sciences, Lund University
Ämne
- Probability Theory and Statistics
Aktiv
Published
Handledare
ISBN/ISSN/Övrigt
- ISBN: 978-91-90202-203
- ISBN: 978-91-90202-210
Försvarsdatum
9 september 2026
Försvarstid
13:00
Försvarsplats
Lecture Hall MH:Hörmander, Centre of Mathematical Sciences, Märkesbacken 4, Faculty of Engineering LTH, Lund University, Lund.
Opponent
- Gideon Amir (Prof.)